Uncorrelatedness in growing networks with preferential survival of nodes

The emergence of uncorrelated growing networks is proved when nodes are removed either uniformly or under the preferential survival rule recently observed in the World Wide Web evolution. To this aim, the rate equation for the joint probability of degrees is derived, and stationary symmetrical solutions are obtained, by passing to the continuum limit. When a uniformly random removal of extant nodes and linear preferential attachment of new nodes are at work, we prove that the only stationary solution corresponds to uncorrelated networks for any removal rate $r\in (0,1)$. In the more general case of preferential survival of nodes, uncorrelated solutions are also obtained. These results generalize the uncorrelatedness displayed by the (undirected) Barabási-Albert network model to models with uniformly random and selective (against low degrees) removal of nodes.

Figure 1

Time evolution of the mean degree [$\stackrel{̲}{k}(t)]$) and the average degree of deleted nodes [$\stackrel{̲}{k^d}(t)$] in growing networks with LPA and PS for $\alpha =0.5$ (left), $1$ (middle), and $2$ (right). The final time is attained when the network size equals ${10}^6$. The values of the parameters are $m=10$ and $r=0.1$. Solid lines correspond to the predicted value of $\stackrel{̲}{k}$ according to Eq. (14): $\stackrel{̲}{k}=18.67$ (left), $18.97$ (middle), and $19.30$ (right). Open circles correspond to the values of $\stackrel{̲}{k^d}(t)$ according to Eq. (1) using the observed values of $\stackrel{̲}{k}(t)$.